Let's go through each question.

1. Question 1: "The set $\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\}$ is a basis of $\mathbb{R}^3$."

   To determine if this set is a basis of $\mathbb{R}^3$, we need to check if the vectors are linearly independent and span $\mathbb{R}^3$.

   • Since there are exactly three vectors in this set and they are in $\mathbb{R}^3$, if they are linearly independent, they will also span $\mathbb{R}^3$.
   • To check linear independence, we can set up the matrix with these vectors as columns and row-reduce it: $\begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix}$
   • If this matrix has a pivot in every row, then the vectors are linearly independent.

   Upon performing row reduction, we find that the matrix does indeed have a pivot in every row, confirming that the vectors are linearly independent. Therefore, this set is a basis for $\mathbb{R}^3$.

   Answer: True

2. Question 2: "The set $\left\{ \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}, \begin{bmatrix} -5 \\ 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ 3 \\ 0 \end{bmatrix} \right\}$ is a basis of $\mathbb{R}^4$."

   For a set to be a basis of $\mathbb{R}^4$, it must have exactly four linearly independent vectors. However, this set has five vectors, which means it cannot be linearly independent in $\mathbb{R}^4$ (since we only need four vectors to span $\mathbb{R}^4$). Therefore, this set cannot be a basis of $\mathbb{R}^4$.

   Answer: False

3. Question 3: "If $A$ is invertible, then the set of vectors made of the columns of $A$ is linearly independent."

   An invertible matrix $A$ has the property that its columns form a linearly independent set. This is because an invertible matrix has a pivot in every row (and column), ensuring that no column can be written as a linear combination of the others. Therefore, if $A$ is invertible, its columns are indeed linearly independent.

   Answer: True

Would you like more details on any of these solutions, or do you have any questions?

Related Questions:

1. What conditions must a set of vectors satisfy to be a basis for a vector space?
2. How can you determine if a set of vectors is linearly independent?
3. Why does having more than four vectors in $\mathbb{R}^4$ prevent a set from being a basis?
4. What is the significance of a matrix being invertible?
5. How does row reduction help in determining linear independence?

Tip:

Remember that a basis for a space like $\mathbb{R}^n$ requires exactly $n$ linearly independent vectors. If you have fewer than $n$ vectors, they can't span the space; if you have more, they can't all be independent.