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

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Published on 4 Oct 2020
KEY WORDS: isomorphism, invertible linear transformation, change of coordinate matrix
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:VWbe 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:VWbe 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:VWbe 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?)
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