Math Problem Statement

what does it mean to write a vector as a linear combination of other vectors

Solution

Writing a vector as a linear combination of other vectors means expressing that vector as the sum of scalar multiples of the other vectors.

Formal Definition:

If you have vectors v1,v2,,vn\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n and you want to express a vector v\mathbf{v} as a linear combination of these vectors, you write:

v=c1v1+c2v2++cnvn\mathbf{v} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_n \mathbf{v}_n

Here:

  • c1,c2,,cnc_1, c_2, \dots, c_n are scalars (often called coefficients).
  • v1,v2,,vn\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n are the vectors you're combining.
  • v\mathbf{v} is the vector you're expressing as the sum of these scaled vectors.

In essence, you're constructing v\mathbf{v} by scaling the vectors v1,v2,,vn\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n by the appropriate scalars and then adding them together.

Example:

Suppose you have vectors v1=(12)\mathbf{v}_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix} and v2=(34)\mathbf{v}_2 = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, and you want to express the vector v=(56)\mathbf{v} = \begin{pmatrix} 5 \\ 6 \end{pmatrix} as a linear combination of v1\mathbf{v}_1 and v2\mathbf{v}_2. You would solve for c1c_1 and c2c_2 in the equation:

(56)=c1(12)+c2(34)\begin{pmatrix} 5 \\ 6 \end{pmatrix} = c_1 \begin{pmatrix} 1 \\ 2 \end{pmatrix} + c_2 \begin{pmatrix} 3 \\ 4 \end{pmatrix}

This results in a system of equations: 5=c11+c235 = c_1 \cdot 1 + c_2 \cdot 3 6=c12+c246 = c_1 \cdot 2 + c_2 \cdot 4

By solving this system, you can find the values of c1c_1 and c2c_2.

Key Points:

  • The scalars c1,c2,,cnc_1, c_2, \dots, c_n can be any real numbers (or even complex numbers in some cases).
  • The number of vectors you use depends on the dimension of the space you're working in. For example, in 3D space, you can represent any vector as a linear combination of three non-collinear vectors.
  • The idea is that any vector within a given span (the space spanned by the vectors) can be written as a combination of the vectors that span the space.

Let me know if you'd like further details or an example on how to solve for the coefficients.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Linear Combinations

Formulas

v = c1 * v1 + c2 * v2 + ... + cn * vn

Theorems

Linear Combination Theorem

Suitable Grade Level

Grades 10-12