6 linear algebra questions
Math 220 Summer Written Homework 4 1. ([C]: option to use a computer to reduce a matrix, but make sure to reference what you used). Let T : R3 → R3 be a linear transformation such that 1 −1 1 3 1 2 T 0 = 5 , T 1 = 0 , and T 1 = −1 . 0 1 0 1 1 4 2 4. Find T −1 2. Assume that each map T given below is a linear transformation. Find the standard matrix of T . After you find the standard matrix, use it to compute T (1, 1). (a) T : R2 → R2 where T (⃗e1 ) = (2, −1) and T (⃗e2 ) = (3, 6). (b) T : R2 → R2 is the transformation that that first reflects points over the origin, then projects onto the x-axis. • Hint. First figure T (e1 ) and T (e2 ), then use these to make the standard matrix of T . Don’t use matrix products yet (that comes later in this homework). You can figure out T (e1 ) and T (e2 ) by tracking where the vectors go visually/geometrically. But you may also do it by computing the appropriate matrix-vector products. 3. Answer the following questions for the linear transformations below. (a) Find the standard matrix. (b) State the domain and codomain. (c) Identify the rank and nullity of the linear transformation. Then give a geometric description of the range and null space of the linear transformation (a *blank* dimensional space living in *blank*). (d) Determine if the transformation is one-to-one, onto, both, or neither. Make sure to use part (c) to justify your answer. x−y x I. T = −y y y x x + y II. S y = z z 4. Provide a description of how the geometric linear transformation T acts on any vector x x= in R2 . For example, an answer could look like “T : R2 → R2 that first reflects y points over the origin, then projects onto the x-axis.” 1 3 0 x (a) T (x) = 0 5 y 0 −1 x (b) T (x) = −1 0 y 5 0 x (c) T (x) = 0 0 y 5. Suppose: 2 −1 • S = TA ◦ TB where S(1, 0) = and S(0, 1) = 3 −1 1 0 • P = TB ◦ TA where P (1, 0) = and P (0, 1) = . 2 5 Note. this question can be done very quickly with little to no computation if you are applying the correct concepts/definitions. (a) What is the standard matrix for S? What is the standard matrix for P ? hint: don’t overthink this, there is nothing to compute here, it should take less than 10 seconds. (b) What does the product AB equal? A and B are the standard matrices for TA and TB . (c) What does the product BA equal? 6. In each of the following: Let TA : R2 → R2 be the linear transformation that rotates and TB : R2 → R2 be the linear transformation that vectors counter-clockwise by 3π 2 reflects vectors over the y-axis. Let A and B be the respective standard matrices. (a) How does TB ◦ TA act on vectors geometrically? How does TA ◦ TB act on vectors geometrically? (you’re just expressing in words what is happening to the vectors and in what order). (b) Find the standard matrix for TB ◦TA by computing the appropriate matrix product (you first need to recall what the standard matrices A,B are for TA and TB ). (c) Find the standard matrix for TA ◦ TB by computing the appropriate matrix product. 1 (d) Set ⃗v = . Confirm that (BA)⃗v = B(A⃗v ) by computing each side separately. 2 2
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