Linear algebra-linear combination

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In summary: Since the last row is "0 0 0" the answer is "yes, every third column is a linear combination of the first two columns".In summary, the conversation is about determining if the third column of a matrix can be written as a linear combination of the first two columns. The attempt at a solution involves using an augmented matrix to solve for the values of a and b, which are then used in the equation x = a(U1) + b(U2). The expert explains that this is the same as asking if x+2y=3, 7a+8y=9, 4a+5b=6 can be solved for a and b, and that the answer is yes since the

Homework Statement

For each matrix, can you write the third column of the matrix as a linear combination
of the first two columns?

$$\left[ \begin{array}{cccc} 1 & 2 & 3 \\ 7 & 8 & 9 \\ 4 & 5 & 6 \end{array} \right]$$

x=a(U1)+b(U2)

The Attempt at a Solution

I let x equal the third column, U1 as the first column, and U2 as the second column. I solved the augmented matrix and got:

$$\left[ \begin{array}{cccc} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array} \right]$$

which a=-1, b=2.

This where I'm confused. Do I just multiply a by the first column and b by the second then that will give me a matrix that is the linear combination wrt the third column?

A linear combination of x and y is a*x + b*y. So you want to know if you can find some a and b such that
a*first column + b*second column is equal to third column.

Can you?

Saying "third column is a linear combination of the first two columns" is the same as saying x+ 2y= 3, 7a+ 8y= 9, 4a+ 5b= 6 for some a, b, c. Can you solve those three equations? One way to solve a system of equations is to set up the "augmented" matrix and row-reduce. Do you see that you have already done that? What are x and y?

By the way, if the question was really "can you write the third column of the matrix as a linear combination
of the first two columns?" then you should have been known the answer as soon as you saw the last row.

1. What is a linear combination in linear algebra?

A linear combination in linear algebra is a mathematical operation performed on a set of vectors, where each vector is multiplied by a scalar and then added together. This operation is often used to create new vectors that can represent different relationships between the original vectors.

2. How is a linear combination represented mathematically?

A linear combination can be represented mathematically using Greek letters, such as alpha and beta, to represent the scalars. The vectors are typically represented using bold letters, and the operation is written as a sum with the scalars multiplied by the vectors, such as alpha * vector1 + beta * vector2.

3. What is the significance of linear combinations in linear algebra?

Linear combinations are significant in linear algebra because they can be used to create new vectors that can represent different relationships between the original vectors. This allows for a deeper understanding and analysis of vector spaces and their properties.

4. Can a linear combination be applied to more than two vectors?

Yes, a linear combination can be applied to any number of vectors. This allows for the creation of even more complex relationships between the vectors and can be helpful in solving real-world problems in fields such as physics, engineering, and economics.

5. How are linear combinations used in solving systems of linear equations?

Linear combinations are used in solving systems of linear equations by creating new equations that represent different linear combinations of the original equations. This can help in finding solutions to the system of equations or determining if the system has no solution or infinitely many solutions.