Let v_i = (v_i(1), v_i(2), v_i(3), v_i(4)), for i=1,2,3,4, be four vectors in R^4 such that ∑_{i=1}^{4} v_i(j) = 0, for each j=1,2,3,4. Let W be the subspace of R^4 spanned by {v_1... Let v_i = (v_i(1), v_i(2), v_i(3), v_i(4)), for i=1,2,3,4, be four vectors in R^4 such that ∑_{i=1}^{4} v_i(j) = 0, for each j=1,2,3,4. Let W be the subspace of R^4 spanned by {v_1, v_2, v_3, v_4}. Then the dimension of W over R is always. (a) either equal to 1 or equal to 4 (b) less than or equal to 3 (c) greater than or equal to 2 (d) either equal to 0 or equal to 4.
Understand the Problem
The question is asking about the dimension of a subspace W in R^4 spanned by four vectors, given a specific condition that their components sum to zero. We need to analyze how this condition affects the possible dimensions of the subspace.
Answer
3
Answer for screen readers
The dimension of the subspace $W$ in $\mathbb{R}^4$ is 3.
Steps to Solve
- Identify the Condition of the Vectors
The vectors in the subspace $W$ in $\mathbb{R}^4$ have components that sum to zero. This means if we have a vector $\mathbf{v} = [v_1, v_2, v_3, v_4]$, the condition is $v_1 + v_2 + v_3 + v_4 = 0$.
- Determine the Dimension of the Subspace
We note that the condition $v_1 + v_2 + v_3 + v_4 = 0$ represents one linear constraint on the four components of the vectors. In general, a single linear constraint reduces the dimension of the space by one. Therefore, we start with $\mathbb{R}^4$, which has a dimension of 4, and we subtract 1 for the constraint. Thus, the dimension of the subspace is:
$$ \text{Dimension of } W = 4 - 1 = 3 $$
- Confirm Spanning with the Base Vectors
Now, we need to confirm that we can construct three linearly independent vectors that satisfy our condition. A possible set of spanning vectors is:
- $\mathbf{v_1} = [1, -1, 0, 0]$
- $\mathbf{v_2} = [1, 0, -1, 0]$
- $\mathbf{v_3} = [1, 0, 0, -1]$
Each of these vectors satisfies the condition $v_1 + v_2 + v_3 + v_4 = 0$.
- Conclude the Dimension
Since we have found three linearly independent vectors that span the subspace $W$, we can conclude that the dimension of $W$ is indeed 3.
The dimension of the subspace $W$ in $\mathbb{R}^4$ is 3.
More Information
In $\mathbb{R}^n$, when $n$ vectors satisfy a single linear constraint, the dimension of the resulting subspace will be $n - 1$. This concept is pivotal in understanding the behavior of vector spaces under constraints, which often appears in various fields such as physics and engineering.
Tips
- Confusing the original dimension with the resulting dimension after applying constraints. Remember that each independent constraint reduces the dimension by one.
- Assuming that all four vectors are needed to span the space without confirming their independence and the effect of the constraint.
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