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MATH 261  Final Exam  02.06.2001

(F : a field, V : vector space over F, ℝ : the set of real numbers) I. True or false?

− − − 1) The matrix [ \begin{matrix}1 & 0 & 0 & b \\ 0 & 1 & −1 & a \\ 0 & 0 & 0 & 0 \end{matrix} ] is row-reduced

− − − 2) W : vector space over F, T : V → W : linear transformation. ⟹ rank (T)+ nullity (T) = dim V.

− − − 3) Let F = ℤ_{⊭}, 𝕎 = \{[ \begin{matrix}x & −x \\ y & z \end{matrix} ] |↶, ↷, 𝟋 ∈ 𝔽\} then

[ \begin{matrix}1 & 1 \\ 0 & 0 \end{matrix} ] ∈ W.

− − − 4) W_{1}, W_{2}, W_{1} ∪ W_{2} : subspaces of V ⇒ W_{1} ⊆ W_{2} or W_{2} ⊆ W_{1}. − − − 5) dimV < ∞, W : vector space over F with dimW < ∞. Then

it is possible to find an isomorphism between V and W.

II. Complete in the blanks with correct statements.

1) If A ∈ F^{n×n}, then A is invertible ⟺ AX = Y has a solution X.............

2) If A ∈ 𝔽^{⋗×⋉}, then the domain of the function A is ....................

3) If W_{1}, W_{2} are subspace of ℝ^{⋬} such that dimW_{1}^{0} = 3, dimW_{2} = 2, and dim(X_{1} ∪ W_{2}) = 1, then dim(W_{1} + W_{2}) =...................

III Choose the corect answer and justify your choice.

1) Let 0 ≠ u ∈ F. Put D_{i}(u) = 1 + (u − 1)e_{ii}. Is D_{i}(u) invertible or not? (e_{ii} is the matrix in whuch the entry in (i, i)-position is 1 and elsewhere 0.)

( ) not invertible ( ) invertible

Justification:

2) Is [ \begin{matrix}1 & 0 & 2 & −1 \\ 0 & 2 & 2 & 2 \\ 0 & 0 & −2 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} ] invertible, or not? If invertible find A^{−1}.

( ) invertible ( ) not invertible

Justification:

3) Let B ∈ F^{n×n} be fixed, V = F^{n×n}, T : V → V A → AB − BA

Is T a linear transformation?

( ) Yes, it is, ( ) No, it is not.

Justification:

4) Let dimV = n, T : V → V : linear operator, R_{T} = N_{T} . Is n even or odd?

( ) even ( ) odd

Justification:

5) Let T be the linear operator on ℝ^{⊯}, the matrix of which in the standard

ordered basis is A = [ \begin{matrix}1 & 2 & 1 \\ 0 & 1 & 1 \\ −1 & 3 & 4 \end{matrix} ] . Find b such that

T (x_{1}, x_{2}, 2) = (x_{1} + 2x_{2} + b, x_{2} + 2, −x_{1} + 3x_{2} + 8).

( ) b = 2, ( ) b = −2 ( ) b = 8, ( ) None Justification:

6) Find the sum of all images of x = −2ϵ_{1} + 3ϵ_{2} + ϵ_{3} under the basis vectors of W^{0} corresponding to α_{1}, α_{2}, α_{3}, where W is spanned by α_{1}, α_{2}, α_{3} with

α_{1} = −ϵ_{1} + 2ϵ_{2} + ϵ_{3}, α_{2} = −2ϵ_{1} + ϵ_{2}, α_{3} = ϵ_{1} + ϵ_{2} + ϵ_{3} in ℝ^{⊯} (ϵ_{i} are the standard ordered basis vectors.) ( ) the sum is 2 ( ) the sum is 1 ( ) None Justification:

IV. Prove: Let S be a linearly independent subset of V. Suppose that β is a vector in V which is not in the subspace spanned by S. Then the set obtained by adjoining β to S is linearly independent.

Proof:

V. Determine all values of the constants a and b for which the system

x_{1} + x_{2} − 2x_{3} = 4
3x_{1} + 5x_{2} − 4x_{3} = 16
2x_{1} + 3x_{2} − ax_{3} = b

has no solution in ℝ

Solution: