Exam (elaborations) linear algebra

National University of Singapore
Department of Mathematics

Semester 2, 2018/19 MA1101R Linear Algebra I Tutorial 11

Questions of this tutorial sheet will be discussed in the tutorial classes in Week 13
(April 15–April 19, 2019).
You are advised to revise Sections 6.4-7.3 before attempting the questions.


Questions

1. Let Q(x1 , x2 , x3 ) = (x1 − x̄)2 + (x2 − x̄)2 + (x3 − x̄)2 where x̄ = 31 (x1 + x2 + x3 ).
 2
− 13 − 13 x1 

3


(a) Show that Q(x1 , x2 , x3 ) = x1 x2 x3 − 1 2 − 1 x2 .

3 3 3
− 13 − 13 2 x3
3
(b) Rewrite Q(x1 , x2 , x3 ) in the form as described in Discussion 6.4.4. (Read also
Example 6.4.5.)


2. Which of the followings are linear transformations? Write down the standard
matrix for each of the linear transformations.
     
2 2 x x x
(a) T1 : R → R such that T1 = for ∈ R2 .
y 2x y
     
2 2 x 2 x
(b) T2 : R → R such that T2 = for ∈ R2 .
y 1 y
     
x 0 x
3 3
(c) T3 : R → R such that T3   y  = y − x for y  ∈ R3 .
  
z y−z z
(d) T4 : Rn → R such that T4 (x) = xT Ay for x ∈ Rn where A = (aij )n×n is a
constant matrix and y = (y1 , y2 , . . . , yn )T is a constant vector.
(e) T5 : Rn → R such that T5 (x) = xT Ax for x ∈ Rn where A = (aij )n×n is a
constant matrix.

(In Parts (d) and (e), R is regarded as R1 .)

3. Let T : R3 → R2 be a linear transformation such that
     
1 0 1
T (u) = , T (v) = and T (w) =
0 1 1

where S = {u, v, w} is a basis for R3 .

1

, Suppose we know that
     
1 1 1
[e1 ]S = 0 , [e2 ]S =  0  and [e3 ]S = −2
1 −1 1

where {e1 , e2 , e3 } is the standard basis for R3 .
Find the standard matrix for T and write down the formula of T .

4. Let T : R5 → R4 be a linear transformation with the standard matrix
 
1 2 3 4 5
2 4 2 4 2
 
.
−1 −2 −3 −2 −1

0 0 4 2 4

(a) Find a basis for the range of T.
(b) Find a basis for the kernel of T.
(c) Use this example to verify the Dimension Theorem for Linear Transformation.


5. Let T : Rn → Rm and S : Rm → Rn such that S ◦ T = I where I : Rn → Rn is the
identity mapping.

(a) Show that R(S) = Rn .
(b) Show that Ker(T ) = {0}.
(c) Suppose n = m and A is the standard matrix for T . Find the standard
matrix for S.


6. Suppose T : Rn → Rn is a linear operator such that {T (e1 ), T (e2 ), . . . , T (en )} is
an orthonormal basis for Rn where {e1 , e2 , . . . , en } is the standard basis for Rn .

(a) Let A be the standard matrix for T . Show that A is orthogonal.
(b) If n = 2, what kind(s) of geometric transformations can T be? Justify your
answer.




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