Vector Spaces and Subspaces

  1. Show that the set \(H=\left\{\begin{bmatrix}x_1\\x_2\\0\end{bmatrix}: x_1, x_2 \in \mathbb{R}\right\}\) is a subspace of \(\mathbb{R}^3. \) solution


  2. Show that the set \(K=\left\{\begin{bmatrix}x_1\\x_2\end{bmatrix}: x_1 \geq 0\right\}\) is not a subspace of \(\mathbb{R}^2. \) solution


  3. Show that the set \(H= \left\{\begin{bmatrix}x_1-x_2\\x_2\\x_2-3x_1\end{bmatrix}: x_1, x_2 \in \mathbb{R}\right\}\) is a subspace of \(\mathbb{R}^3 \).   solution


  4. Check if the set \(W=\left\{\left[\begin{array}{l}a \\ b \\ c\end{array}\right]: a+b=c+3 \right\}\) is a subspace of \(\mathbb{R}^{3}\).   solution


  5. Let \(W=\left\{\left[\begin{array}{l}a \\ b \\ c\end{array}\right]: a-c=0, 2b=3c \right\}.\) Show that \(W\) is a subspace of \(\mathbb{R}^{3}\).   solution


  6. Determine if the set \(H = \left\{\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right]: a, b \text{ are integers}\right\}\) is a subspace of \(\mathbb{M}_{2 \times 2}\), the vector space of all \(2\times 2\) matrices.  solution


  7. Check if the set \(H= \left\{\begin{bmatrix}3s\\s+2\end{bmatrix}: s \in \mathbb{R}\right\}\) is a subspace of \(\mathbb{R}^2\).    solution


  8. Check if the set \(H = \{ax^3+bx^2+cx+d:a,b,c,d \in \mathbb{Z}\}\), that is, the set of all polynomials of degree at most 3 with integer coefficients, is a subspace of \(\mathbb{P}_n \), the vector space of all polynomials of degree at most \(n\).   solution


More examples