---
title: "LACS.PAR.7: Vector Space Applications (Georgia Mathematics standard)"
code: LACS.PAR.7
cite_as: Georgia Standards of Excellence, Mathematics, LACS.PAR.7
subject: math
grades:
  - high-school
language: en
canonical_url: https://georgiahomeroom.org/standards/code/LACS.PAR.7
md_url: https://georgiahomeroom.org/standards/code/LACS.PAR.7.md
alternate_language_url: https://georgiahomeroom.org/es/standards/code/LACS.PAR.7.md
source_url: https://case.georgiastandards.org/ims/case/v1p1/CFDocuments/e9dd7229-3558-4df2-85c6-57b8938f6180
source_last_changed: 2024-10-09
data_built: 2026-09-18
publisher: Georgia Homeroom, a free public-interest project of Georgia Civic Data. Not affiliated with or endorsed by the Georgia Department of Education or any school district.
up: https://georgiahomeroom.org/standards/math/high-school.md
index: https://georgiahomeroom.org/standards/index.md
---

# LACS.PAR.7: Vector Space Applications

Solve contextual, mathematical problems using vector spaces to explain real-life phenomena.

- Subject: Mathematics
- Appears in: High School, https://georgiahomeroom.org/standards/math/high-school.md

## Expectations

- Use properties of vector spaces to solve problems.
  - **LACS.PAR.7.1**: Determine whether a given set of vectors generates a vector space.
  - **LACS.PAR.7.2**: Justify whether a subset of a vector space is a subspace.
  - **LACS.PAR.7.3**: Determine whether a given vector is in the linear span of a set of vectors.
  - **LACS.PAR.7.4**: Determine whether two vector subspaces are orthogonal; find the orthogonal component of a given subspace.
- Find and use vectors as a basis.
  - **LACS.PAR.7.5**: Determine whether a set of vectors is a basis for a vector space.
  - **LACS.PAR.7.6**: Find the dimension of a vector space; find the dimensions of the row space, column space, and kernel for a given matrix; find the rank of a matrix.
  - **LACS.PAR.7.7**: Find a matrix representing a linear map.
  - **LACS.PAR.7.8**: Determine the change of representation for a linear transformation given two different bases on a vector space.
  - **LACS.PAR.7.9**: Determine if two matrices are similar; determine if two matrices are orthogonal.
  - **LACS.PAR.7.10**: Find an orthogonal basis for a given basis or subspace by applying the Gram-Schmidt orthonormalization process.
  - **LACS.PAR.7.11**: Perform QR factorization of a matrix to solve matrix equations.
- Apply vector spaces and vectors as a basis in programming to solve contextual problems.
  - **LACS.PAR.7.12**: Apply the method of least squares to find the line or parabola of best fit to approximate data in context.
  - **LACS.PAR.7.13**: Apply the grow-and-shrink algorithm in the minimum spanning forest problem in GF(2).
  - **LACS.PAR.7.14**: Apply the Exchange Lemma to image perspective rendering.
  - **LACS.PAR.7.15**: Use bases to represent images and sounds as wavelets; perform wavelet transformation, implementation, and decomposition through programming.
  - **LACS.PAR.7.16**: Program a Fast Fourier Transform to store a sequence of amplitude samples.
  - **LACS.PAR.7.17**: Apply the Rank Theorem to demonstrate the simple authentication scheme.

## Georgia Milestones coverage

No Georgia Milestones blueprint cites this standard.

Source: Georgia Standards of Excellence, published by the Georgia Department of Education at case.georgiastandards.org.