Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 16

MATH152 April 2016

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Question A 16
Consider the matrix representation $A$ of a linear transformation $T:\mathbb {R} ^{5}\rightarrow \mathbb {R} ^{3}$. Circle all correct answers below: (a) $A$ is invertible. (b) $A$ has three rows. (c) $A$ has three columns. (d) $T({\textbf {x}})={\textbf {0}}$ for some ${\textbf {x}}\neq {\textbf {0}}$. (e) $A=A^{T}$.

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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Review the properties of linear transformation.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Since the linear transformation is from $\mathbb {R} ^{5}\to \mathbb {R} ^{3},$ we have $A:=A_{3\times 5}.$ (a) False. A is not a square matrix, so it is not invertible. (b) True. A has three rows. (c) False. A has $5$ columns. (d) True. The dimension of the domain is $5$, and is larger than the dimension of codomain which is 3. (e) False. Transpose of A is a $5\times 3$ matrix and is not the same as A. Therefore, the correct ones are $\color {blue}(b),(d)$.