Question:
If \(W_1,W_2\) are two subspaces of a vector space \(V(F)\), then \(L(W_1\cup W_2)=\)
If \(W_1,W_2\) are two subspaces of a vector space \(V(F)\), then \(L(W_1\cup W_2)=\)
Show Hint
The span of the union of two subspaces is equal to the sum of the two subspaces.
Updated On: Jul 3, 2026
- \(W_1\cap W_2\)
- \(W_1\cup W_2\)
- \(W_1+W_2\)
- \(W_1-W_2\)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Concept:
Here \(L(W_1\cup W_2)\) means the linear span of the set \(W_1\cup W_2\). The sum of two subspaces is defined as \[ W_1+W_2=\{w_1+w_2:w_1\in W_1,\ w_2\in W_2\} \]
Step 1: Understand span of union. The span of \[ W_1\cup W_2 \] contains all finite linear combinations of vectors from \(W_1\) and \(W_2\). Therefore, every vector in the span can be written as \[ w_1+w_2 \] where \[ w_1\in W_1,\qquad w_2\in W_2 \]
Step 2: Compare with sum of subspaces. This is exactly the definition of \[ W_1+W_2 \] Therefore, \[ L(W_1\cup W_2)=W_1+W_2 \]
Step 3: Final answer. \[ \boxed{W_1+W_2} \]
Here \(L(W_1\cup W_2)\) means the linear span of the set \(W_1\cup W_2\). The sum of two subspaces is defined as \[ W_1+W_2=\{w_1+w_2:w_1\in W_1,\ w_2\in W_2\} \]
Step 1: Understand span of union. The span of \[ W_1\cup W_2 \] contains all finite linear combinations of vectors from \(W_1\) and \(W_2\). Therefore, every vector in the span can be written as \[ w_1+w_2 \] where \[ w_1\in W_1,\qquad w_2\in W_2 \]
Step 2: Compare with sum of subspaces. This is exactly the definition of \[ W_1+W_2 \] Therefore, \[ L(W_1\cup W_2)=W_1+W_2 \]
Step 3: Final answer. \[ \boxed{W_1+W_2} \]
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