Question:
If $T:R^{3}\rightarrow R^{2}$ be a linear transformation defined by $T(a,b,c)=(a+b+c,a-b-c)$, then which of the following is an element in the null space of T.
If $T:R^{3}\rightarrow R^{2}$ be a linear transformation defined by $T(a,b,c)=(a+b+c,a-b-c)$, then which of the following is an element in the null space of T.
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Null space elements are the "invisible" inputs that get crushed to zero by the transformation.
Updated On: Jul 3, 2026
- (1,-1,0)
- (0,1,1)
- (0,1,-1)
- (1,1,1)
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The Correct Option is C
Solution and Explanation
Step 1: Concept
An element is in the null space if $T(a,b,c) = (0,0)$.
Step 2: Meaning
We need to solve: $a+b+c = 0$ $a-b-c = 0$
Step 3: Analysis
Adding the two equations: $2a = 0 \Rightarrow a = 0$. Substituting $a=0$ into the first equation: $b+c = 0 \Rightarrow b = -c$.
Step 4: Conclusion
Any vector of the form $(0, b, -b)$ is in the null space. Setting $b=1$ gives the vector (0, 1, -1). Final Answer: (C)
An element is in the null space if $T(a,b,c) = (0,0)$.
Step 2: Meaning
We need to solve: $a+b+c = 0$ $a-b-c = 0$
Step 3: Analysis
Adding the two equations: $2a = 0 \Rightarrow a = 0$. Substituting $a=0$ into the first equation: $b+c = 0 \Rightarrow b = -c$.
Step 4: Conclusion
Any vector of the form $(0, b, -b)$ is in the null space. Setting $b=1$ gives the vector (0, 1, -1). Final Answer: (C)
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