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Section 3.1 Properties of Linear Transformations

Definition 3.1.1.

A linear transformation is a function \(f: \mathbb{L}_1 \longrightarrow \mathbb{L}_2\text{,}\) such that for any \(\mathbf{P}\text{,}\) \(\mathbf{Q} \in \mathbb{L}_1\text{,}\) and \(c \in \mathbb{R}\text{:}\)

\begin{equation*} f(\mathbf{P}+\mathbf{Q}) = f(\mathbf{P})+f(\mathbf{Q}) \text{ and } f(c\mathbf{P}) = cf(\mathbf{P}) \end{equation*}

  1. \(\displaystyle f(\mathbf{P}+\mathbf{Q}) = f(\mathbf{P})+f(\mathbf{Q})\)

  2. \(\displaystyle f(c\mathbf{P}) = cf(\mathbf{P})\)

If \(T:\mathbb{L}_1 \longrightarrow \mathbb{L}_2\) is a linear transformation explain why:

  1. \(\displaystyle T(\mathbf{0}) = \mathbf{0}\)

  2. \(T( -\mathbf{A}) = -T(\mathbf{A})\) for any \(\mathbf{A}\in \mathbb{L}_1\)

  3. \(T(\mathbf{A}-\mathbf{B}) = T(\mathbf{A})-T(\mathbf{B})\) for any \(\mathbf{A}\text{,}\) \(\mathbf{B} \in \mathbb{L}_1\)

Which of the following are functions?

  1. \(f:\mathbb{Q}\) \(\rightarrow \mathbb{Q}\) defined by \(f(\frac{a}{b})= ab\)

  2. \(g:\mathbb{Q}\) \(\rightarrow \mathbb{Q}\) defined by \(g(\frac{a}{b}) = \frac{b}{a}\)

  3. \(h:\mathbb{Q}^2\) \(\rightarrow \mathbb{Q}\) defined by \(h(\frac{a}{b},\frac{c}{d}) = \frac{ac}{bd}\)

  4. \(j:\mathbb{Q}\) \(\rightarrow \mathbb{Q}^2\) defined by \(j(\frac{a}{b}) = (a,b)\)

  5. \(k:\mathbb{Q}\) \(\rightarrow \mathbb{Q}\) defined by \(k(\frac{a}{b}) = \frac{a^2}{b^2}\)

Which of the following functions are linear transformations?

  1. \(f:\mathbb{R}\) \(\rightarrow \mathbb{R}\) defined by \(f(x)= x^2\)

  2. \(g:\mathbb{R}\) \(\rightarrow \mathbb{R}\) defined by \(g(x) = 3x+1\)

  3. \(h:\mathbb{R}^3\) \(\rightarrow \mathbb{R}^2\) defined by \(h(x_1,x_2,x_3) = (x_1,x_2)\)

  4. \(j:\mathbb{P}\) \(\rightarrow \mathbb{R}\) defined by \(j(p(x)) = p(1)\)

  5. \(k:C[0,1]\rightarrow C[0,1]\) defined by \(k(f(x)) = f\ '(x)\)

Which of the following functions are \(1-1\text{?}\)

  1. \(f:\mathbb{R}^2\) \(\rightarrow \mathbb{R}^2\) defined by \(f(x_1,x_2)= (x_2 ,x_1)\)

  2. \(g:\mathbb{R}^2\) \(\rightarrow \mathbb{R}^2\) defined by \(g(x_1, x_2) = (x_1,0)\)

  3. \(h:C[0,1] \rightarrow C[0,1]\) defined by \(h(f) = \int^x_0f\)

  4. \(j:C[0,1] \rightarrow \mathbb{R}\) defined by \(j(f) = \int^1_0f\)

  5. \(k:\mathbb{P}\) \(\rightarrow \mathbb{P}\) defined by \(k(p(x)) = x\cdot p(x)\)

Which of the following functions are onto?

  1. \(f:\mathbb{R}^2\) \(\rightarrow \mathbb{R}^2\) defined by \(f(x_1,x_2)= (x_2 ,x_1)\)

  2. \(g:\mathbb{R}^2\) \(\rightarrow \mathbb{R}^2\) defined by \(g(x_1, x_2) = (x_1,0)\)

  3. \(h:C[0,1] \rightarrow C[0,1]\) defined by \(h(f) = \int^x_0f\)

  4. \(j:C[0,1] \rightarrow \mathbb{R}\) defined by \(j(f) = \int^1_0f\)

  5. \(k:\mathbb{P}\) \(\rightarrow \mathbb{P}\) defined by \(k(p(x)) = x\cdot p(x)\)

For a function to be \(1-1\) it must have the property that if you put in two different inputs, \(x \neq y\text{,}\) then their outputs are different, \(f(x) \neq f(y)\text{.}\) Recall that the contrapositive of an implication is equivalent to that implication. Use the contrapositive to rewrite what it means for a function for be \(1-1\text{.}\) What does it mean for a function to be onto?

Show

  1. \(f:\mathbb{R}^2 \longrightarrow \mathbb{R}^3\) defined by \(f(x_1,x_2)= (x_1,x_1+x_2,x_1-x_2)\) is \(1-1\)

  2. \(g:\mathbb{R}^4 \longrightarrow \mathbb{R}^2\) defined by \(g(x_1,x_2,x_3,x_4) = (x_1 \cdot x_2, x_3\cdot x_4)\) is onto