Try it yourself 1 1 5 2 − 1 1 Are the vectors , , linearly independent? 3 1 9 1 1 1 Put them in a matrix and row reduce! 1 2 3 1 1 − 1 1 1 → → 1 − 1 1 1 1 2 3 1 5 1 9 1 5 1 9 1 1 − 1 1 1 1 − 1 1 1 → 0 3 2 − 1 0 3 2 − 1 0 6 4 − 4 0 0 0 − 2 The final echelon matrix has no zero row, so the original rows are linearly independent.
A has r rows and c columns; A : R c → R r Columns below have equivalent conditions (except in parethesis) A x = 0 implies x = 0 A x = b has solutions for any b pivot in every column pivot in every row columns linearly independent rows linearly independent rows span all of R c columns span all of R r hits all of R r . distinct points to distinct points (can only happen if c ≤ r ) (can only happen if r ≤ c ) If A is square, i.e. r = c , there’s a pivot in every row if and only if there’s a pivot in every column so these are all equivalent.
Scalar multiplication
Scalar multiplication Just like for vectors, multiplying a matrix by a scalar just means multiplying every element of the matrix by that scalar.
Scalar multiplication Just like for vectors, multiplying a matrix by a scalar just means multiplying every element of the matrix by that scalar. � 1 � 3 � � 2 6 3 · = 3 4 9 12
Scalar multiplication Just like for vectors, multiplying a matrix by a scalar just means multiplying every element of the matrix by that scalar. � 1 � 3 � � 2 6 3 · = 3 4 9 12 � − 2 � � � 1 − 1 0 2 0 − 2 · = − 2 3 1 4 − 6 − 2
Scalar multiplication Just like for vectors, multiplying a matrix by a scalar just means multiplying every element of the matrix by that scalar. � 1 � 3 � � 2 6 3 · = 3 4 9 12 � − 2 � � � 1 − 1 0 2 0 − 2 · = − 2 3 1 4 − 6 − 2 � 2 � 0 3 5 � 0 0 � 0 · = 7 11 13 0 0 0
Matrix addition And you can add matrices of the same size by adding them termwise.
Matrix addition And you can add matrices of the same size by adding them termwise. � 1 � 2 � 3 � � � − 1 0 4 1 3 1 + = 0 3 4 1 2 4 1 5 8
Matrix addition And you can add matrices of the same size by adding them termwise. � 1 � 2 � 3 � � � − 1 0 4 1 3 1 + = 0 3 4 1 2 4 1 5 8 1 2 4 3 5 5 + = 0 3 2 − 1 2 2 1 4 3 0 4 4
Matrix addition And you can add matrices of the same size by adding them termwise. � 1 � 2 � 3 � � � − 1 0 4 1 3 1 + = 0 3 4 1 2 4 1 5 8 1 2 4 3 5 5 + = 0 3 2 − 1 2 2 1 4 3 0 4 4 � 1 1 2 � − 1 0 + 0 3 0 3 4 1 4
Matrix addition And you can add matrices of the same size by adding them termwise. � 1 � 2 � 3 � � � − 1 0 4 1 3 1 + = 0 3 4 1 2 4 1 5 8 1 2 4 3 5 5 + = 0 3 2 − 1 2 2 1 4 3 0 4 4 � 1 1 2 � − 1 0 + 0 3 they’re not the same size 0 3 4 1 4
Matrix transpose
Matrix transpose The transpose of the matrix is what you get by reflecting along a northwest-southeast diagonal.
Matrix transpose The transpose of the matrix is what you get by reflecting along a northwest-southeast diagonal. This makes the old first column
Matrix transpose The transpose of the matrix is what you get by reflecting along a northwest-southeast diagonal. This makes the old first column into the new first row, etcetera.
Matrix transpose The transpose of the matrix is what you get by reflecting along a northwest-southeast diagonal. This makes the old first column into the new first row, etcetera. T � 2 2 4 � 1 − 1 1 3 = 4 3 8 − 1 8
Matrix transpose The transpose of the matrix is what you get by reflecting along a northwest-southeast diagonal. This makes the old first column into the new first row, etcetera. T � 2 2 4 � 1 − 1 1 3 = 4 3 8 − 1 8 T 1 � � 2 = 1 2 3 3
Matrix transpose The transpose of the matrix is what you get by reflecting along a northwest-southeast diagonal. This makes the old first column into the new first row, etcetera. T � 2 2 4 � 1 − 1 1 3 = 4 3 8 − 1 8 T 1 � � 2 = 1 2 3 3 � 1 � 1 � T � 3 3 = 3 2 3 2
Matrix entries We write M i , j or M ij for the entry in the i ′ th row and j ′ th column of the matrix M .
Matrix entries We write M i , j or M ij for the entry in the i ′ th row and j ′ th column of the matrix M . � 3 � 3 1 A = 1 5 8 A 1 , 1 = 3 , A 2 , 3 = 8
Matrix multiplication Given two matrices, A , B ,
Matrix multiplication Given two matrices, A , B , if A has as many columns as B has rows,
Matrix multiplication Given two matrices, A , B , if A has as many columns as B has rows, then there is a matrix product AB .
Matrix multiplication Given two matrices, A , B , if A has as many columns as B has rows, then there is a matrix product AB . The matrix product is determined by the formula ( AB ) ij = A i 1 B 1 j + A i 2 B 2 j + · · · + A in B nj where n is the number of columns of A , or of rows of B .
Matrix multiplication Given two matrices, A , B , if A has as many columns as B has rows, then there is a matrix product AB . The matrix product is determined by the formula ( AB ) ij = A i 1 B 1 j + A i 2 B 2 j + · · · + A in B nj where n is the number of columns of A , or of rows of B . AB has as many rows as A
Matrix multiplication Given two matrices, A , B , if A has as many columns as B has rows, then there is a matrix product AB . The matrix product is determined by the formula ( AB ) ij = A i 1 B 1 j + A i 2 B 2 j + · · · + A in B nj where n is the number of columns of A , or of rows of B . AB has as many rows as A and as many columns as B .
Matrix multiplication 1 0 1 2 1 2 ? ? 2 1 − 1 0 1 3 = ? ? 1 2 2 3 0 1 ? ? 0 3
Matrix multiplication 1 0 1 2 1 2 6 2 1 − 1 0 1 3 = 1 2 2 3 0 1 0 3 1 × 1 + 2 × 2 + 1 × 1 + 2 × 0 = 6
Matrix multiplication 1 0 1 2 1 2 6 2 1 − 1 0 1 3 = 0 1 2 2 3 0 1 0 3 − 1 × 1 + 0 × 2 + 1 × 1 + 3 × 0 = 0
Matrix multiplication 1 0 1 2 1 2 6 2 1 − 1 0 1 3 = 0 1 2 2 3 0 1 8 0 3 2 × 1 + 3 × 2 + 0 × 1 + 1 × 0 = 8
Matrix multiplication 1 0 1 2 1 2 6 10 2 1 − 1 0 1 3 = 0 1 2 2 3 0 1 8 0 3 1 × 0 + 2 × 1 + 1 × 2 + 2 × 3 = 10
Matrix multiplication 1 0 1 2 1 2 6 10 2 1 − 1 0 1 3 = 0 11 1 2 2 3 0 1 8 0 3 − 1 × 0 + 0 × 1 + 1 × 2 + 3 × 3 = 11
Matrix multiplication 1 0 1 2 1 2 6 10 2 1 − 1 0 1 3 = 0 11 1 2 2 3 0 1 8 6 0 3 2 × 0 + 3 × 1 + 0 × 2 + 1 × 3 = 6
Matrix multiplication Another way to see it: Write the second matrix as a “row of columns” � � B = b 1 b 2 · · · b n Then: � � � � AB = A b 1 b 2 · · · b n = A b 1 A b 2 · · · A b n The matrix-vector product is a special case.
Try it yourself � � 3 � 1 2 1 � = ? − 1 1 1 2 � 3 � � � 1 1 2 = ? 1 2 − 1 1
Matrix multiplication The matrix product is associative,
Matrix multiplication The matrix product is associative, distributes over matrix addition,
Matrix multiplication The matrix product is associative, distributes over matrix addition, but is not generally commutative.
Matrix multiplication The matrix product is associative, distributes over matrix addition, but is not generally commutative. Indeed, the dimensions of A and B can be such that AB makes sense but BA does not;
Matrix multiplication The matrix product is associative, distributes over matrix addition, but is not generally commutative. Indeed, the dimensions of A and B can be such that AB makes sense but BA does not; and we saw on the last slide that even if they both make sense, they need not be equal.
Try it yourself � 1 � � � 0 1 − 1 0 = ? 0 1 − 2 3 1 � 1 0 0 � 1 − 1 0 = 0 1 0 ? − 2 3 1 0 0 1
The identity matrix � 1 � � � � � 0 1 − 1 0 1 − 1 0 = 0 1 − 2 3 1 − 2 3 1 � 1 0 0 � � � 1 − 1 0 1 − 1 0 = 0 1 0 − 2 3 1 − 2 3 1 0 0 1
The identity matrix We write I n for the matrix with 1’s along the diagonal, and zeroes everywhere else.
The identity matrix We write I n for the matrix with 1’s along the diagonal, and zeroes everywhere else. � 1 1 0 0 � 0 I 1 = [1] , I 2 = , I 3 = 0 1 0 0 1 0 0 1
The identity matrix We write I n for the matrix with 1’s along the diagonal, and zeroes everywhere else. � 1 1 0 0 � 0 I 1 = [1] , I 2 = , I 3 = 0 1 0 0 1 0 0 1 If I n · M is defined, i.e., M has n rows, then I n · M = M
The identity matrix We write I n for the matrix with 1’s along the diagonal, and zeroes everywhere else. � 1 1 0 0 � 0 I 1 = [1] , I 2 = , I 3 = 0 1 0 0 1 0 0 1 If I n · M is defined, i.e., M has n rows, then I n · M = M If M · I m is defined, i.e., M has m columns, then M · I m = M
The identity matrix We write I n for the matrix with 1’s along the diagonal, and zeroes everywhere else. � 1 1 0 0 � 0 I 1 = [1] , I 2 = , I 3 = 0 1 0 0 1 0 0 1 Note that the identity matrix I n is the unique reduced echelon n × n matrix with a pivot in every row (or equivalently, every column).
Try it yourself 1 0 0 1 2 3 = ? 0 1 0 4 5 6 0 1 1 7 8 9 0 1 0 1 2 3 = ? 1 0 0 4 5 6 0 0 1 7 8 9 1 0 0 1 2 3 = ? 0 2 0 4 5 6 0 0 1 7 8 9
Try it yourself 1 0 0 1 2 3 1 2 3 = 0 1 0 4 5 6 4 5 6 0 1 1 7 8 9 11 13 15 0 1 0 1 2 3 4 5 6 = 1 0 0 4 5 6 1 2 3 0 0 1 7 8 9 7 8 9 1 0 0 1 2 3 1 2 3 = 0 2 0 4 5 6 8 10 12 0 0 1 7 8 9 7 8 9
Elementary row operations You can do an elementary row operation
Elementary row operations You can do an elementary row operation by multiplying on the left
Elementary row operations You can do an elementary row operation by multiplying on the left by the matrix which is obtained by performing that row operation on the identity matrix.
Try it yourself! � 1 � � � 2 3 − 2 = ? 1 3 − 1 1 � � 1 � � 3 − 2 2 = ? − 1 1 1 3
Try it yourself! � 1 � 1 � � � � 2 3 − 2 0 = 1 3 − 1 1 0 1 � � 1 � 1 � � � 3 − 2 2 0 = − 1 1 1 3 0 1
Matrix inversion If A is a matrix, we say A is invertible
Matrix inversion If A is a matrix, we say A is invertible if there is some other matrix B such that BA and AB are both identity matrices.
Matrix inversion If A is a matrix, we say A is invertible if there is some other matrix B such that BA and AB are both identity matrices. In this case we say that B is the inverse of A , and write it as A − 1 .
Matrix inversion If A is a matrix, we say A is invertible if there is some other matrix B such that BA and AB are both identity matrices. In this case we say that B is the inverse of A , and write it as A − 1 . The inverse is unique if it exists:
Matrix inversion If A is a matrix, we say A is invertible if there is some other matrix B such that BA and AB are both identity matrices. In this case we say that B is the inverse of A , and write it as A − 1 . The inverse is unique if it exists: if BA = I and AC = I
Matrix inversion If A is a matrix, we say A is invertible if there is some other matrix B such that BA and AB are both identity matrices. In this case we say that B is the inverse of A , and write it as A − 1 . The inverse is unique if it exists: if BA = I and AC = I then B = BI = B ( AC ) = ( BA ) C = IC = C
Try it yourself! Is the identity matrix invertible?
Try it yourself! Is the identity matrix invertible? yes I n · I n = I n
Matrix inversion If A is an invertible matrix, then the following equations are equivalent: x = A − 1 b A x = b
Matrix inversion If A is an invertible matrix, then the following equations are equivalent: x = A − 1 b A x = b In particular,
Matrix inversion If A is an invertible matrix, then the following equations are equivalent: x = A − 1 b A x = b In particular, A x = 0 has only the zero solution x = A − 1 0 = 0
Matrix inversion If A is an invertible matrix, then the following equations are equivalent: x = A − 1 b A x = b In particular, A x = 0 has only the zero solution x = A − 1 0 = 0 For any b , the equation A x = b has the solution x = A − 1 b .
A has r rows and c columns; A : R c → R r Columns below have equivalent conditions (except in parethesis) A x = 0 implies x = 0 A x = b has solutions for any b pivot in every column pivot in every row columns linearly independent rows linearly independent rows span all of R c columns span all of R r one-to-one onto (can only happen if c ≤ r ) (can only happen if r ≤ c ) If A is square, i.e. r = c , there’s a pivot in every row if and only if there’s a pivot in every column so these are all equivalent.
Matrix inversion If A is an invertible matrix,
Matrix inversion If A is an invertible matrix, then A is square,
Matrix inversion If A is an invertible matrix, then A is square, and all the conditions on the previous slide hold.
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