Math Problem Statement

Show that the given vector is in the span of S: a)  1 0 0 1 , S =  1 0 −1 1 ,  0 1 0 1 ,  1 1 0 0.

Solution

To show that the vector v=(1001)\mathbf{v} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} is in the span of the set S={(1011),(0101),(1100)}S = \left\{ \begin{pmatrix} 1 \\ 0 \\ -1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix} \right\}, we need to determine if there exist scalars c1,c2,c3c_1, c_2, c_3 such that:

c1(1011)+c2(0101)+c3(1100)=(1001)c_1 \begin{pmatrix} 1 \\ 0 \\ -1 \\ 1 \end{pmatrix} + c_2 \begin{pmatrix} 0 \\ 1 \\ 0 \\ 1 \end{pmatrix} + c_3 \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}

This gives the following system of equations:

c_1(1) + c_2(0) + c_3(1) &= 1 \quad \text{(from the first row)} \\ c_1(0) + c_2(1) + c_3(1) &= 0 \quad \text{(from the second row)} \\ c_1(-1) + c_2(0) + c_3(0) &= 0 \quad \text{(from the third row)} \\ c_1(1) + c_2(1) + c_3(0) &= 1 \quad \text{(from the fourth row)} \end{aligned}$$ We now solve this system of equations: ### Step 1: Simplify the system From the first equation: $$c_1 + c_3 = 1 \tag{1}$$ From the second equation: $$c_2 + c_3 = 0 \tag{2}$$ From the third equation: $$- c_1 = 0 \quad \Rightarrow \quad c_1 = 0 \tag{3}$$ From the fourth equation: $$c_1 + c_2 = 1 \quad \Rightarrow \quad 0 + c_2 = 1 \quad \Rightarrow \quad c_2 = 1 \tag{4}$$ ### Step 2: Solve for $$c_3$$ From equation (2), we have $$c_2 + c_3 = 0$$, so: $$1 + c_3 = 0 \quad \Rightarrow \quad c_3 = -1$$ ### Step 3: Conclusion The solution is $$c_1 = 0$$, $$c_2 = 1$$, and $$c_3 = -1$$. Thus, we can express $$\mathbf{v}$$ as a linear combination of the vectors in $$S$$: $$\mathbf{v} = 0 \begin{pmatrix} 1 \\ 0 \\ -1 \\ 1 \end{pmatrix} + 1 \begin{pmatrix} 0 \\ 1 \\ 0 \\ 1 \end{pmatrix} + (-1) \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix}$$ Therefore, the vector $$\mathbf{v} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}$$ is in the span of $$S$$. Do you want more details on any step? Or further explanation? Here are 5 questions that build on this problem: 1. What does it mean for a vector to be in the span of a set of vectors? 2. Can a set of vectors span all of $$\mathbb{R}^n$$? Under what conditions? 3. How do you determine if a set of vectors is linearly independent? 4. What is the geometric interpretation of the span of two vectors in $$\mathbb{R}^2$$? 5. How would the solution change if we added more vectors to $$S$$? **Tip**: When solving systems of equations derived from vector spans, always check for consistency in each row to ensure the system has a solution.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Linear Combinations

Formulas

c1*v1 + c2*v2 + c3*v3 = v

Theorems

Span of a Set of Vectors

Suitable Grade Level

Undergraduate Level