MATH 54

Midterm 1 | Spring '16| Wehrheim
1a) Rewrite the system of linear equations \begin{align} x_2 - x_4 &= 5 \\ x_1 + x_3 &= -1 \\ x_1 + x_2 + x_3 + x_4 &= 0 \end{align}
in terms of a linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$, a vector b in $\mathbb{R}^p$, and an unknown x in $\mathbb{R}^q$. In particular, specify $T$ and b explicitly. (This involves a choice of appropriate integers $m, n, p, q$.)

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1b) Find a basis for the kernel of $T$. (That is, find linear independent vector(s) which span kernel($T$).)

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1c) Find the solution set for the system in 1a) \begin{align} x_2 - x_4 &= 5 \\ x_1 + x_3 &= -1 \\ x_1 + x_2 + x_3 + x_4 &= 0 \end{align}
and write it in parametric vector form.
(You can use any method but should show your work or explain your reasoning.)

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2) Let $T: V \to W$ be a linear transformation between vector spaces $V, W$, and suppose that $w \ne 0$ in $W$ is a vector for which $T(x) = w$ has exactly one solution $x$. What does this imply about kernel($T$)? Prove your statement without appealing to theorems or solution principles from book or lecture.

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3) Let $\mathbb{P}$ be the vector space of polynomials and suppose that $T: \mathbb{P} \to \mathbb{R}^3$ is a linear transformation so that $T(1 + t^2) = \left[ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right], \quad T(t^2) = \left[ \begin{array}{c} -1 \\ 0 \\ 4 \end{array} \right], \quad T(t^3) = \left[ \begin{array}{c} 3 \\ 2 \\ 1 \end{array} \right]$ Use this information to calculate $T(1 + 2t^2 + 3t^3)$.
(Hint: The functions $p_1(t) = 1 + t^2, p_2(t) = t^2, p_3(t) = t^3, q(t) = 1 + 2t^2 + 3t^3$ above are vectors in $\mathbb{P}$. However, there is no need to use this notation.)

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4a) Consider the functions $t^2 - 1, t^3$, and $\frac{1}{t^2 + 1}$ as vectors in $\mathcal{C}$, the vector space of continuous functions of $-\infty < t < \infty$. State the definition of what it would mean for these three vectors to be linearly independent in $\mathcal{C}$.

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4b) Using your definition from 4a), show that $t^2 - 1, t^3$, and $\frac{1}{t^2 + 1}$ are linearly independent in $\mathcal{C}$. (Hint: plug in convenient values of $t$.)

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5) Make a list of all possible reduced echelon forms of matrices $A$ with $2$ rows and $3$ columns, so that the corresponding linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^2, \mathbf{x} \to A\mathbf{x}$ is onto.

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6a) Give an example of two matrices $A \ne 0$ and $B \ne 0$ whose product is $AB = 0$.

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6b) Someone claims they found an example of $AB = 0$ where $A$ is invertible, and $B$ is invertible, too. Prove them wrong!

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7) Calculate the determinant of $A = \left[ \begin{array}{c c c c} 2 & 0 & 0 & 0 \\ 1 & 5 & 0 & 0 \\ 8 & 6 & 4 & 0 \\ 6 & 5 & 4 & 3 \end{array} \right]$ by showing each step in a cofactor expansion.

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8a) For which parameter(s) $h$ is $A = \left[ \begin{array}{c c c} 6 & h & 3 \\ 0 & 2 & 0 \\ -1 & 0 & 0 \end{array} \right]$ invertible?
For those parameter(s), find the inverse.

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8b) What matrix multiplication would you compute to check your result in 8a)?
(Full credit is given for just writing down the matrices to multiply - but some calculation is recommended.)

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