MATH 51TA Section Notes for Tue 14 Oct 08Jason LoToday’s topics:• linear transformations• matrix multiplicationExample 1 Consider the two functionsT : R3→ R4:(x, y, z) 7→ (x, y + z, 0, xyz)S : R4→ R4:(a, b, c, d) 7→ (b, cd, a + c, a + b)1. Are T, S linear transformations? (Ans: No.) Why or why not?2. Write down a formula for S ◦ T . Is it a linear transformation? Why orwhy not?S ◦ T : R3→ R4: (x, y, z) 7→ (y + z, 0, x, x + y + z)And it is a linear transformation (can check that both conditions for alinear transformation are satisfied).3. Write down the matrix for S ◦T .The matrix corresponding to the linear transformation S ◦ T isA = [(S ◦ T )(e1))|(S ◦ T )(e2)|(S ◦ T )(e3)]=0 1 10 0 01 0 01 1 14. Find ker (S ◦ T )1By definition,ker (S ◦ T ) = {x ∈ R3: S ◦ T (x) = 0}= {xyz∈ R3: y + z = 0, x = 0, x + y + z = 0}= {xyz∈ R3: x = 0, y = −z}= {xyz∈ R3:xyz= z0−11for some z ∈ R= span0−11Note that since the linear transformation S ◦T corresponds to the matrixA, we have ker (S◦T ) = N (A), the null space of A, and it is 1-dimensional.5. What is dim (im (S ◦ T ))?We have im (S ◦ T ) = C(A), and ker (S ◦ T ) = N (A). We also knowdim N(A) + dim C(A) = number of columns of A= 3So dim (im (S ◦ T )) = 3 − 1 = 2.Example 2. (Ex 14.15)Example 3. Let A be the matrix for Rot−π4: R2→ R2, rotation counter-clockwise by the angle −π/4 radians in R2.1. Find A, A2, A4, A8.We have A =cos θ −sin θsin θ cos θwhere θ = −π/4. That is, A =√2/2√2/2−√2/2√2/2.We can either compute A2directly as the product A2= AA, or note thatA2corresponds to applying Rot−π4twice, i.e. rotating counterclockwise bythe angle −π/2, and conclude A2=cos −π/2 −sin −π/2sin −π/2 cos −π/2=0 1−1 0.A4is the matrix for rotation counterclockwise by π.A8is the matrix for rotation counterclockwise by (−π/4) ·8 = −2π, underwhich nothing gets moved. So A8should be the identity matrix.2. Let B be the the matrix for Rotπ4: R2→ R2. Find AB,
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