[3]

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 $m, n, p, q$.)

**1a)**Rewrite the system of linear equationsin 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 $m, n, p, q$.)

[7]

**1b)**Find the solution set of the system in 1a) and write it in parametric vector form.[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.)[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.[10]

using only definitions and the following information (no theorems etc.):

1.) is a linear transformation,

2.) kernel $T$ = span

3.)

**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.)

[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 $\mathbb{P}_2$.[7]

**4b)**Using your definition from 4a), show that $t^2, 1 + t$, and $(1 + t)^2$ do span $\mathbb{P}_2$.[6]

**5a)**Determine the standard matrix associated to the linear transformation

[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 $T(x) = b$.[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.)[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.[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[4]

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