[Merged by Bors] - feat(Analysis/InnerProductSpace): Gram matrix det ≠ 0 iff input vectors are independent

[wwylele]

Small lemma extracted from #37295. Convenient for talking about the determinant directly without detour through Pos(Semi)def

---

[github-actions]

PR summary ade87df682

Import changes exceeding 2%

%	File
+14.41%	Mathlib.Analysis.InnerProductSpace.GramMatrix

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Analysis.InnerProductSpace.GramMatrix	2145	2454	+309 (+14.41%)

Import changes for all files

Files	Import difference
Mathlib.Analysis.InnerProductSpace.GramMatrix	309

---

Declarations diff

+ PosSemidef.posDef_iff_det_ne_zero
+ det_gram_ne_zero_iff_linearIndependent
+ linearIndependent_of_det_gram_ne_zero

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[j-loreaux]

I'm okay with the import increase since this is a leaf (or at least it has no downstream import increases).

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• ci (staging) / Post-CI job
• ci (staging) / Build
• ci (staging) / Lint style
• ci (staging) / Post-Build Step