A) The complex numbers is a 2-dimensional R-vector space. B) The set of invertible 2 × 2 matrices with entries from R is a subspace of M2(R). C) If T : V → W is a linear transformation, then it is always true that T (0V ) = 0W

A) The complex numbers is a 2-dimensional R-vector space. B) The set of invertible 2 × 2 matrices with entries from R is a subspace of M2(R). C) If T : V → W is a linear transformation, then it is always true that T (0V ) = 0W

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

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Question

Answer True or False

A) The complex numbers is a 2-dimensional R-vector space.

B) The set of invertible 2 × 2 matrices with entries from R is a subspace

of M₂(R).

C) If T : V → W is a linear transformation, then it is always true that

T (0_V ) = 0_W

D) The zero vector is always an eigenvector for a linear transformation.

E) If X ⊕ Y = X ⊕ Z then Y = Z.

F) If a function f : X → Y is surjective, then there exists g : Y → X such that f ◦ g = id_Y .

G) If U and W are subspaces of V, then U ∪ W is a subspace of V.

H) ⟨T,v⟩ is always a T-invariant subspace of V ( v ∈ V ).

I) For any linear transformation T : V → V , ker(T ) ≤ ker(T²).

Combination of a real number and an imaginary number. They are numbers of the form a + b , where a and b are real numbers and i is an imaginary unit. Complex numbers are an extended idea of one-dimensional number line to two-dimensional complex plane.

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