Determine if the Vector is in the Column Space

Problem

a=[[1],[3],[1],[2]],b=[[−2],[−4],[−4],[−3]],x=a+b

Solution

1. Define the column space. The column space of a set of vectors is the set of all possible linear combinations of those vectors.

2. Calculate the vector x by performing vector addition.

x=[[1],[3],[1],[2]]+[[−2],[−4],[−4],[−3]]

x=[[1+(−2)],[3+(−4)],[1+(−4)],[2+(−3)]]

x=[[−1],[−1],[−3],[−1]]

3. Determine if x is in the column space of {a,b} A vector x is in the column space if there exist scalars (c_1) and (c_2) such that (c_1)*a+(c_2)*b=x

4. Identify the coefficients. Since x was defined as the sum of a and b we have (c_1)=1 and (c_2)=1

1*a+1*b=x

5. Conclude that because x is a linear combination of a and b it must be in the column space spanned by those vectors.

Final Answer

x∈Col{a,b}

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