Question:
If $W=\{\left(\begin{matrix}x& y\\ z& 0\end{matrix}\right):x,y,z\in R\}$ is a subspace of the vector space $M_{2}$ of $2\times2$ matrices over the field of real numbers R, then $dimW=_$
If $W=\{\left(\begin{matrix}x& y\\ z& 0\end{matrix}\right):x,y,z\in R\}$ is a subspace of the vector space $M_{2}$ of $2\times2$ matrices over the field of real numbers R, then $dimW=_$
Show Hint
Dimension = Number of arbitrary constants in the general form of the vector.
Updated On: Jul 3, 2026
- 1
- 2
- 3
- 4
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Concept
The dimension of a subspace is the number of free variables or basis vectors needed to represent any element in that space.
Step 2: Meaning
Any matrix in $W$ can be written as: $x\left(\begin{matrix}1& 0
0& 0\end{matrix}\right) + y\left(\begin{matrix}0& 1
0& 0\end{matrix}\right) + z\left(\begin{matrix}0& 0
1& 0\end{matrix}\right)$.
Step 3: Analysis
There are three independent parameters ($x, y, z$) that define the set, and the three matrices shown above form a basis for $W$.
Step 4: Conclusion
Since there are three vectors in the basis, the dimension is 3. Final Answer: (C)
The dimension of a subspace is the number of free variables or basis vectors needed to represent any element in that space.
Step 2: Meaning
Any matrix in $W$ can be written as: $x\left(\begin{matrix}1& 0
0& 0\end{matrix}\right) + y\left(\begin{matrix}0& 1
0& 0\end{matrix}\right) + z\left(\begin{matrix}0& 0
1& 0\end{matrix}\right)$.
Step 3: Analysis
There are three independent parameters ($x, y, z$) that define the set, and the three matrices shown above form a basis for $W$.
Step 4: Conclusion
Since there are three vectors in the basis, the dimension is 3. Final Answer: (C)
Was this answer helpful?
0
0
Top AP ECET BSc Mathematics Mathematics Questions
- The integrating factor of $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is}
- If $y = y(x)$ is the solution of $\left(\frac{2 + \sin x}{y + 1}\right) \frac{dy}{dx} + \cos x = 0$ with $y(0) = 1$, then $y\left(\frac{\pi}{2}\right) = $
- The solution of $y^{4}dx + 2xy^{3}dy = \frac{ydx - xdy}{x^{3}y^{3}}$ is}
- Which of the following is an integrating factor of $(x^{2} + y^{2} + 2x)dx + 2ydy = 0$?
- The differential equation $\frac{dy}{dx} = \frac{-(x + x^{8} + py^{2})}{y^{8} - y + qxy}$ is exact if}
View More Questions
Top AP ECET BSc Mathematics Algebra Questions
- On the set of integers Z, the operation defined as following is not a binary operation.
- In the group $(R-\{-1\}, *)$ where $*$ is defined by $a*b = a+b+ab, \forall a,b \in R-\{-1\}$, inverse of the element 2 is}
- The intersection of the subgroups 2Z, 3Z and 5Z, i.e., $2Z \cap 3Z \cap 5Z$ of (Z, +) is
- The permutations $f = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 4 & 8 & 2 & 5 & 1 & 3 & 7 & 6 \end{pmatrix}, g = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 5 & 1 & 2 & 8 & 7 & 4 & 3 & 6 \end{pmatrix}$ of $S_8$ are respectively}
- A noncyclic group among the following is
View More Questions
Top AP ECET BSc Mathematics Questions
- The integrating factor of $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is}
- If $y = y(x)$ is the solution of $\left(\frac{2 + \sin x}{y + 1}\right) \frac{dy}{dx} + \cos x = 0$ with $y(0) = 1$, then $y\left(\frac{\pi}{2}\right) = $
- The solution of $y^{4}dx + 2xy^{3}dy = \frac{ydx - xdy}{x^{3}y^{3}}$ is}
- Which of the following is an integrating factor of $(x^{2} + y^{2} + 2x)dx + 2ydy = 0$?
- The differential equation $\frac{dy}{dx} = \frac{-(x + x^{8} + py^{2})}{y^{8} - y + qxy}$ is exact if}
View More Questions