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Linear Algebra Exercises: Finding Ker(T), Linear Independence, Basis and Dimension, Assignments of Algebra

Exercises on linear algebra, focusing on finding the kernel of a linear transformation t, determining linear independence, finding a basis and calculating the dimension of a subspace. The exercises involve finding the matrix representation of a linear transformation and applying it to vectors, as well as determining if given matrices form a linearly independent set and finding a basis for the span of those matrices.

Typology: Assignments

Pre 2010

Uploaded on 08/16/2009

koofers-user-6f4
koofers-user-6f4 🇺🇸

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Math 233
Complete the following exercises.
1. Find ker(T) of T:M2×2 M2×2is the linear transformation defined by
T a b
c d != c2a2b
0ad+c!.
2. Consider the set of 2 ×2 matrices
S=( 1 1
0 0 !, 0 1
0 1 !, 1 1
0 1 !, 3 2
0 0 !).
(a) Is Sa linearly independent set?
(b) Find a basis Bfor span(S).
(c) What is the dimension of span(S)?
(d) Write the matrix A= 6 2
0 1 !as a linear combination of the vectors in the
basis B.
(e) What is the coordinate vector [A]B?
(f) For what value of kis [A]BRk?
(g) Let Cbe the standard basis of M2×2. For what value of kis [A]CRk? What is
[A]C?
3. Let Vbe a subspace of Pof polynomials of the form p(x) = ax4+bx3+cx2+dx +e
such that ac= 0 and e3b= 0.
(a) Find a basis for V.
(b) What is the dimension of V?
1

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Math 233

Complete the following exercises.

  1. Find ker(T ) of T : M 2 × 2 −→ M 2 × 2 is the linear transformation defined by

T

a b c d

c − 2 a 2 b 0 a − d + c

  1. Consider the set of 2 × 2 matrices

S =

(a) Is S a linearly independent set? (b) Find a basis B for span(S). (c) What is the dimension of span(S)?

(d) Write the matrix A =

as a linear combination of the vectors in the basis B. (e) What is the coordinate vector [A]B? (f) For what value of k is [A]B ∈ Rk? (g) Let C be the standard basis of M 2 × 2. For what value of k is [A]C ∈ Rk? What is [A]C?

  1. Let V be a subspace of P of polynomials of the form p(x) = ax^4 + bx^3 + cx^2 + dx + e such that a − c = 0 and e − 3 b = 0.

(a) Find a basis for V. (b) What is the dimension of V?