University:
Whitman College
Course:
MATH-240 | Linear Algebra
Academic year:

2022
Views:

356

Pages:

1
Author:

Angelica Kent

Some Additions to Review Set 2 1. What are the two ways we defined matrix multiplication, AB? (Hint: One way was the rwo-column rule). 2. Let A, B be given below. Form the matrix product AB, if defined:   −1 1 2 0 −1 A= B= 2 1  1 1 0 1 2 3. Given the matrix A, B below:   1 2 3 A =  −1 1 0  3 2 0   1 −1 2 1 1  B= 2 1 0 1 (a) Compute only the (2, 3) entry of AB: (b) Compute only the (3, 2) entry of AB T : (c) Compute B − 3I3 : (d) Compute C23 for matrix A (that’s the (2,3) cofactor). 4. If A is the 2 × 3 matrix below, find a matrix C so that AC = I, but note that C is not the inverse of A. To simplify your computations, I’ve given you one form for C that you might use.   c11 c12 −1 2 −1 A= C =  c21 c22  6 −9 3 0 0 5. Suppose A is n×n with the property that Ax = ~0 has only the trivial solution. Without using the invertible matrix theorem, explain directly why the equation Ax = b must have a solution for every b. 6. Explain why the columns of A2 span IRn whenever the columns of A are linearly independent. (Hint: You might think about whether or not A2 must be invertible). 7. Suppose subspace H is the span of the two vectors below in set B:     2   1    0 , 3  = {v1 , v2 } B=   0 0 (a) Does B span IR3 ? Why or why not? (b) Find [v1 ]B (c) If c = (3, 3, 0), find [c]B 1

Linear Algebra Some Additions to Review Set 2

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