MATH 54

Midterm 1 | Spring '17 | Wehrheim

[3] 1a) Rewrite the system of linear equations
in terms of a linear transformation , a vector $b$ in and an unknown $x$ in In particular, specify $T$ and $b$ explicitly. (This involves a choice of appropriate integers $m, n, p, q$.)

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[7] 1b) Find the solution set of the system in 1a) and write it in parametric vector form.

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[6] 1c) Find the solution set for the homogeneous system associated to the inhomogeneous system in 1a). (You can use any method but should show your work or explain your reasoning.)

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[6] 2) Let $T : V → W$ be a linear transformation between vector spaces $V, W$. Give the definition of kernel($T$) and state what it is a subspace of. Then prove your statement without appealing to theorems from book or lecture.

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[10] 3) Find the general solution of and explain why there cannot be any other solutions,
using only definitions and the following information (no theorems etc.):
1.) is a linear transformation,
2.) kernel $T$ = span
3.)

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[3] 4a) Consider the functions $t^2, 1 + t$, and $(1 + t)^2$ as vectors in $\mathbb{P}_2$, the vector space of polynomials up to degree $2$. State the definition of what it would mean for these three vectors to span $\mathbb{P}_2$.

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[7] 4b) Using your definition from 4a), show that $t^2, 1 + t$, and $(1 + t)^2$ do span $\mathbb{P}_2$.

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[6] 5a) Determine the standard matrix associated to the linear transformation


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[4] 5b) Explain whether $T$ is one-to-one, onto, both, or neither, by relating these properties to uniqueness and existence of solutions to equations $T(x) = b$.

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[5] 6a) Replace the · below with entries 1, 0, or * (to denote entries that can be any real number) to create a list of all possible 2 × 3 matrices in reduced echelon form. (Hint: There are less than 8 different forms.)

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[5] 6b) State (don’t prove) criteria for reading off from the reduced echelon form of a matrix whether the associated linear transformation is one-to-one or onto. (Hint: You probably already used these in 5b.) Then apply these criteria to label the matrices above with the one-to-one/onto properties that the associated linear transformations have.

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[6] 7a) State a pivot criterion for deciding whether the columns of a matrix are linearly independent. Then apply it to decide whether the vectors are linearly independent in

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[4] 7b) State the definition of being linearly independent, and check this condition.

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