linear algebra
Math 220 World Campus Written Homework 5 Be sure to show all your work. 1 0 −2 1 −1 −1 2 −1 1 4 . 1. Let A = ,B= , and C = −3 −2 3 4 −2 3 2 −3 4 Perform each computation below or explain why the operation is undefined. (a) AB (d) B 2 (b) BA (e) CB T (c) A2 (f) 2C T − B 2. Let T : R2 → R2 be the transformation that that rotates vectors counterclockwise by 3π . 2 (a) Describe how T −1 must act on vectors (just give a brief justification with words and/or drawings). (b) Find the standard matrix for T −1 by plugging in the appropriate θ into the formula for rotation matrices. (c) Confirm, by performing the computations, that AA−1 = I2 and A−1 A = I2 . Here A is the standard matrix for T (that we learned about previously), and A−1 is the matrix you found in (b). 3. Let A, B, and C be n × n where the inverses A−1 , B −1 , C −1 exist. Simplify the expression as much as possible. Show all work for the simplification. (AC −1 )−1 (AC −1 )(AB T )T (CAT )−1 4. The following question are quick concept checks on the material from the video on rank of AT /row space of A. (a) Suppose you have a 5×6 matrix with rank 4. State the dimension of each of the following: the null space, the column space, and the row space. (b) Suppose A is a 6×7 matrix with rank = 4. How many rows and columns does AT have? What is the rank and nullity of AT . (c) Suppose the rank of a 4×3 matrix is 3. Do the rows form a linearly independent or dependent set? (d) Suppose the rank of a 3×4 matrix is 3. Do the rows form a linearly independent or dependent set? (e) What is the largest possible rank of a 3×9 matrix? Of a 7×4 matrix? 1 5. Determine whether the following matrices are singular. Justify your reasoning (you can look at the rank, whether the columns are linearly independent, or whether the rows are linearly independent). 7 9 0 1 −2 3 1 2 4 4 1 (a) −2 3 0 (b) 0 (c) 0 3 1 4 5 0 3 −6 9 0 0 7 6. We saw in the lecture videos that projecting onto the x-axis is a non-invertible linear 1 0 transformation. So, the corresponding standard matrix A = is singular. In 0 0 this question, you are going to give 6 separate arguments on why the matrix is singular. Each each part, the argument should begin with answer the corresponding question, and stating what that implies about invertibility by the invertible matrix theorem. (a) State the rank of A (and what that implies about invertibility by the invertible matrix theorem – you’re doing this for each part of this question). (b) Find a ⃗b so that A⃗x = ⃗b has no solution. (c) Find a ⃗b so that A⃗x = ⃗b has infinitely many solutions. (d) Determine whether the columns are linearly independent or dependent (justify why). (e) Determine whether the rows are linearly independent or dependent (justify why). (f) Does the reduced row echelon form equal I2 ? 2
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