# MATB24H3 Lecture Notes - Lecture 4: Linear Map, Coordinate Vector

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Published on 4 Oct 2020

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MATB24 TUTORIAL PROBLEMS 4, WEEK OF June8-June12

KEY WORDS: isomorphism, invertible linear transformation, change of coordinate matrix

RELEVANT SECTIONS IN THE TEXTBOOK:Sec 3.3, 3.4, 7.1 FB or 3C,3D SA

WARM-UP: As usual, write down a complete deﬁnition or a complete mathematical character-

ization for the following terms.

•An invertible linear transformation

•Give an equivalent condition for a linear transformation being invertible in terms of

injectivity and surjectivity.

•Give an equivalent condition for a linear transformation Tbeing invertible in terms of

the Kernel and image of T.

•Let Bbe a an ordered basis for a vector space W. Deﬁne the B-coordinates of a vector

~v ∈W.

•Let Bbe a an ordered basis for a vector space W. Give an isomorphism between W

and Rdim W

A: In class we said (or will say) a linear transformation respects the structure of a vector

space, for instance it maps a subspace to a subspace, the zero vector to zero vector, and so on.

In this question you investigate how a linear transformation treats a linear independent set and

a spanning set. We use the following result

Lemma 0.1. Let T:V→Wbe a linear transformation.

(1) Tis one-to-one if and only if ker T={0V}.

(2) Tis onto if and only if img(T) = W

(1) (a) Consider I={ex, e2x, e3x}in F.Iis linearly independent (why?). Let V=

Span(I). Let T:V→ F be a linear transformation, and suppose

T(ex) = 1, T (e2x) = cos2x, T (e3x) = sin2x

Write down a formula for Tof an arbitrary element of V

(b) Show that T(I)is not linearly independent.

(c) Prove that Tis not one-to one.1

(d) Show that T(I)is not a spanning set for F.

(e) Prove that Tis not onto.

(2) Let T:V→Wbe a linear transformation. Prove that T(I)is a linearly independent

subset of Wfor every linearly independent subset Iof Vif and only if Tis one to one.2

(3) Let T:V→Wbe a linear transformation. Let Sbe a spanning set for V. Prove Tis

onto if and only of T(S)is a spanning set for W.

(4) Prove that the ﬁnite-dimensional vector spaces Vand Ware isomorphic if and only if

dim(V) = dim(W).

1You can ﬁnd a nonzero vector in the kernel of T

2For every linearly independent subset Iis a key information in one of the two directions (which one?)