Question

which of the following statements are true and which are false ? justify your answer with a short proof or a counterexample. i) r^2 has infinitely many non zero, proper vector subspaces. ii) if t:v -> w is one-one linear transformation between two finite dimensional vector spaces v and w then t is invertible. iii) if a^k = 0 for a square matrix a, then all the eigen values of a are non zero. iv) every unitary operator is invertible. v) every system of homogeneous linear equations has a non zero solution.

Answer 1

Answer on Question #81384 – Math – Linear Algebra

which of the following statements are true and which are false? justify your answer with a short proof or a counterexample.

Question

(i) $\mathbb{R}^2$ has infinitely many nonzero, proper vector subspaces

Solution

Let $\{e_1, e_2\}$ be a basis in $\mathbb{R}^2$. Then

$$W_n = \{\alpha(e_1 + ne_2), \alpha \in \mathbb{R}\}, n \in \mathbb{N}$$

are all different proper vector subspaces. Indeed, suppose $W_n = W_m$. Then

$$e_1 + ne_2 = \alpha(e_1 + me_2)$$

for some $\alpha$.

Then

$$(\alpha - 1)e_1 + (m\alpha - n)e_2 = 0$$

from which $\alpha = 1, \alpha = n/m$, i.e. $n = m$.

This proves that $W_n$ are all different. And there are infinitely many of them.

Conclusion: $\mathbb{R}^2$ has infinitely many nonzero, proper vector subspaces – true.

Question

ii) if $T: V \to W$ is one-one linear transformation between two finite dimensional vector spaces $V$ and $W$ then $T$ is invertible

Solution

Let $T$ be one-to-one. For any $y \in W$ define $Sy$ as the (unique) solution to $Tx = y$.

Then obviously

$$S(Tx) = x.$$

It is sufficient to prove that $S$ is linear.

$$S(\alpha_1y_1 + \alpha_2y_2) = \alpha_1Sy_1 + \alpha_2Sy_2$$

Let $Sy_1 = x_1, Sy_2 = x_2$. Then

$$T(\alpha_1x_1 + \alpha_2x_2) = T(\alpha_1Sy_1 + \alpha_2Sy_2) = \alpha_1T(Sy_1) + \alpha_2T(Sy_2) = \alpha_1y_1 + \alpha_2y_2,$$

from which

$$\alpha_1 x_1 + \alpha_2 x_2 = S(\alpha_1 y_1 + \alpha_2 y_2)$$

which proves linearity.

Conclusion: if $T: V \to W$ is one-one linear transformation between two finite dimensional vector spaces $V$ and $W$ then $T$ is invertible - true.

Question

iii) if $A^k k = 0$ for a square matrix $A$, then all the eigen values of $a$ are nonzero

Solution

Consider $A = 0$. It has all zero eigenvalues.

Conclusion: if $A^k k = 0$ for a square matrix $A$, then all the eigen values of $a$ are nonzero - false.

Question

iv) every unitary operator is invertible

Solution

By definition an unitary operator $U$ is such that

$$U U^* = U^* U,$$

thus $U^*$ is inverse of $U$.

Conclusion: every unitary operator is invertible - true.

Question

v) every system of homogeneous linear equations has a nonzero solution

Solution

Consider a system:

$$\left\{ \begin{array}{l} x_1 + x_2 = 0 \\ x_1 - x_2 = 0 \end{array} \right.$$

It has only zero solution.

Conclusion: every system of homogeneous linear equations has a nonzero solution - false.

Answer:

(i) $\mathbb{R}^2$ has infinitely many non-zero, proper vector subspaces - true;

ii) if $T: V \to W$ is one-one linear transformation between two finite dimensional vector spaces

V and W then T is invertible – true;

iii) if $A^k = 0$ for a square matrix A, then all the eigen values of a are nonzero – false;

iv) every unitary operator is invertible – true;

v) every system of homogeneous linear equations has a nonzero solution – false.

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