Linear Combinations

## Description

We can use basic vector operations in Sage to build a linear combinations of vectors. Consider $\mathbf v_1 = (2, 3, 5)$, $\mathbf v_2 = (7, 11, 13)$, and $\mathbf v_3 = (17, 19, 23)$. The cell below computes the linear combination $\mathbf v_1 + 6 \mathbf v_2 - 2 \mathbf v_3$.

## Sage Cell

#### Code

```
v1 = vector(QQ, [2, 3, 5])
v2 = vector(QQ, [7, 11, 13])
v3 = vector(QQ, [17, 19, 23])
v1 + 6*v2 - 2*v3
```

## Options

#### Linear Combinations of matrices

We also compute linear combinations of matrices. However, `matrix(QQ, [2, 3, 5]) + vector(QQ, [7, 11, 13])` will produce an error.

v2 = matrix(QQ, [7, 11, 13])

#### Code

```
v1 = matrix(QQ, [2, 3, 5])
v2 = matrix(QQ, [7, 11, 13])
v3 = matrix(QQ, [17, 19, 23])
v1 + 6*v2 - 2*v3
```

## Tags

CC:

Primary Tags: Linear Algebra: Euclidean spaces

Secondary Tags: Euclidean spaces: Linear combinations

## Related Cells

- Vectors in Sage. Vectors in Sage.

## Attribute

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Date: 10 Apr 2020 15:51

Submitted by: Zane Corbiere