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• Does an Extra Vector Change the Span? Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set \[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\] still a spanning set for […]
• Compute Power of Matrix If Eigenvalues and Eigenvectors Are Given Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where \[\mathbf{u}=\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} \text{ […]
• Prove that any Set of Vectors Containing the Zero Vector is Linearly Dependent Prove that any set of vectors which contains the zero vector is linearly dependent.   Solution. Let $\mathbf{0}$ be the zero vector, and $\mathbf{v}_1, \cdots, \mathbf{v}_k$ are the other vectors in the set. Then we have the non-trivial linear combination \[1 \cdot […]
• Application of Field Extension to Linear Combination Consider the cubic polynomial $f(x)=x^3-x+1$ in $\Q[x]$. Let $\alpha$ be any real root of $f(x)$. Then prove that $\sqrt{2}$ can not be written as a linear combination of $1, \alpha, \alpha^2$ with coefficients in $\Q$.   Proof. We first prove that the polynomial […]
• Linearly Dependent if and only if a Vector Can be Written as a Linear Combination of Remaining Vectors Let $V$ be a vector space over a scalar field $K$. Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ be the set of vectors in $V$, where $n \geq 2$. Then prove that the set $S$ is linearly dependent if and only if at least one of the vectors in $S$ can be written as […]
• Determine Linearly Independent or Linearly Dependent. Express as a Linear Combination Determine whether the following set of vectors is linearly independent or linearly dependent. If the set is linearly dependent, express one vector in the set as a linear combination of the others. \[\left\{\, \begin{bmatrix} 1 \\ 0 \\ -1 \\ 0 […]
• Vector Space of Polynomials and Coordinate Vectors Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} &p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\ &p_3(x)=2x^2, &p_4(x)=2x^2+x+1. \end{align*} (a) Use the basis […]
• Any Vector is a Linear Combination of Basis Vectors Uniquely Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as \[\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,\] where $c_1, c_2, c_3$ are […]