#### MATH 54

##### Midterm 1 | Spring '17| Wehrheim
 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 $m, n, p, q$.)

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

 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.)

 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.

 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.) 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 $\mathbb{P}_2$.

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

 5a) Determine the standard matrix associated to the linear transformation 5b) Explain whether $T$ is one-to-one, onto, both, or neither, by relating these properties to uniqueness and existence of solutions to equations $T(x) = b$.

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