1. Determine which of the following subsets of are sub-spaces. If so, prove it, if not, specify which conditions of a sub-space are violated.

a. Vectors of the form:

Answer:
This is a subspace.

  • Closure under addition: If and , then , which is still in the set.
  • Closure under scalar multiplication: If and , then , which is still in the set.
  • Contains the zero vector: is in the set.

Thus, this is a subspace.


b. Vectors of the form:

Answer:
This is not a subspace.

  • Does not contain the zero vector: The zero vector is not in the set because the third component is always .
  • Closure under addition: If and , then , which is not in the set.
  • Closure under scalar multiplication: If and , then , which is not in the set unless .

Thus, this is not a subspace.


c. Vectors of the form: where

Answer:
This is a subspace.

  • Closure under addition: If and satisfy and , then satisfies .
  • Closure under scalar multiplication: If satisfies and , then satisfies .
  • Contains the zero vector: satisfies .

Thus, this is a subspace.


d. Vectors of the form: where

Answer:
This is not a subspace.

  • Does not contain the zero vector: The zero vector satisfies , so it is in the set. However, the set fails other conditions.
  • Closure under addition: Let and . Both satisfy , but does not satisfy .
  • Closure under scalar multiplication: Let and . Then does not satisfy .

Thus, this is not a subspace.


e. Vectors of the form: where

Answer:
This is not a subspace.

  • Closure under scalar multiplication: Let and . Then , which is not in the set because .

Thus, this is not a subspace.


f. Vectors of the form: where and

Answer:
This is a subspace.

  • Closure under addition: If and satisfy , , , and , then satisfies and .
  • Closure under scalar multiplication: If satisfies and , and , then satisfies and .
  • Contains the zero vector: satisfies and .

Thus, this is a subspace.


2. Determine if the shaded regions of define a subspace.

Answer:
The images are not provided, so I cannot analyze the shaded regions. Please provide the images or describe the shaded regions.


3. Find the null space of .

a.

Answer:
The null space of is the set of all vectors such that .
We solve the system:

From the first equation, . Substituting into the second equation:

Thus, the null space is:


b.

Answer:
The null space of is the set of all vectors such that .
We solve the system:

From the second equation, . Substituting into the first equation:

Substituting into the third equation:

Thus, the null space is:


4. Check whether or not is in and is in range where with:

Answer:

  • Check if :
    Compute :

Since , .

  • Check if :
    We need to find such that .
    Solve the system:
From the second equation, $x_1 = 1 - 2x_2$. Substituting into the first equation:

(1 - 2x_2) + 3x_2 - x_3 = 5 \implies 1 + x_2 - x_3 = 5 \implies x_2 - x_3 = 4.

Let $x_2 = t$, then $x_3 = t - 4$ and $x_1 = 1 - 2t$. Thus, a solution exists:

\mathbf{x} = \begin{bmatrix}1 - 2t \ t \ t - 4\end{bmatrix}.

Therefore, $\mathbf{v} \in \operatorname{range}(T)$. --- ### 5. Classify the subspaces of $\mathbb{R}^{2}$. **Answer:** The subspaces of $\mathbb{R}^{2}$ are: 1. The zero subspace: $\{\mathbf{0}\}$. 2. Lines through the origin: Any set of the form $\{k\mathbf{v} \mid k \in \mathbb{R}\}$, where $\mathbf{v} \neq \mathbf{0}$ is a fixed vector. 3. The entire space $\mathbb{R}^{2}$. These are the only subspaces of $\mathbb{R}^{2}$.